Research

Our research group has obtained important research results related to:

- selection of simplified models and parameter estimation using limited data

- development of new techniques for parameter estimation in differential equation (DE) models

- development of process models of interest to our industrial sponsors

Research Contributions from 2007-2011

i) Selection of Simplified Models and Parameter Estimation using Limited Data
Developers of fundamental models often have difficulties deciding which parameters they should estimate using available data and which parameters they should leave at initial guesses, or remove via simplification. Estimating too many parameters can lead to numerical problems and poor predictions, especially when data are not very informative. Our group has developed improved estimability algorithms for ranking model parameters based on the sensitivity of predicted responses to parameters, uncertainties in measured responses and initial parameter guesses, and correlated effects of parameters (Thompson et al., 2009). Until recently, it was difficult to know how many parameters to estimate from the ranked list, without resorting to arbitrary cut-off values (Yao et al., 2003) or brute-force cross-validation (Thompson et al., 2009). With this problem in mind, Wu et al. (2007) developed a critical ratio Rc and used it to confirm that simplified models (SMs) with fewer parameters often give better predictions than “extended” models (EMs). We used Rc to analyze and compare commonly-used model selection criteria (MSC), e.g., the Akaike Information Criterion and the Bayesian Information Criterion (Wu et al., 2011a) and developed a new MSC based on the critical ratio Rc (Wu et al., 2011b). We then showed that this criterion can be used to select the optimal number of parameters to estimate from the ranked list produced by estimability analysis (Wu et al., 2011c). The improved estimability ranking method and new MSC are being used and tested when estimating parameters in industrial models that we are developing for our sponsors.

ii) Parameter Estimation in Differential Equation Models
Two approaches were developed for estimating parameters in DE models using basis functions (e.g., B-splines) to make computations more tractable and robust: i) a Generalized Smoothing (GS) approach that uses a “parameter cascade” (Ramsay et al., 2007), and ii) and an Approximate Maximum Likelihood Estimation (AMLE) approach that explicitly considers both measurement noise and stochastic process disturbances (Varziri et al., 2008 a-d). In the GS approach, an outer optimization loop estimates the parameters in the ODE model, while the inner loop computes spline coefficients that are treated as nuisance parameters. AMLE uses a single objective function to simultaneously determine model parameters and spline coefficients. These new techniques can accommodate unmeasured states, data available at non-uniform time intervals, and unknown or poorly-known initial conditions. AMLE was extended to deal with non-stationary disturbances (Varziri et al., 2008b), and approximate confidence-interval expressions were obtained for AMLE and GS based on linearization. Use of AMLE and GS were illustrated using a simple continuous reactor model and a more complicated nylon reactor model. In our ongoing research, we will extend the GS and AMLE methodologies for use with more-complex models and we will develop inference techniques to properly account for nonlinearity.

Research Projects for 2010-2012

Our ongoing research is aimed at:
i) Developing novel methods to obtain parameter estimates for nonlinear stochastic differential equation (SDE) models using basis functions to simplify the estimation problem.
ii) Developing techniques to obtain inference regions for parameters and predictions that properly account for nonlinearity in SDE models.
iii) Developing techniques for sequential experimental design for fundamental nonlinear models in chemical processes based on estimability analysis
iv) Testing and developing advanced multivariate statistical techniques for use in soft sensors.
v) Developing new models of chemical processes that satisfy the needs of our industrial sponsors and that provide test problems for parameter estimation techniques.